Theoretical and Mathematical Physics

, Volume 198, Issue 3, pp 425–454 | Cite as

Quantum Mechanical Equivalence of the Metrics of a Centrally Symmetric Gravitational Field

  • M. V. Gorbatenko
  • V. P. NeznamovEmail author


We analyze the quantum mechanical equivalence of the metrics of a centrally symmetric uncharged gravitational field. We consider the static Schwarzschild metric in spherical and isotropic coordinates, stationary Eddington–Finkelstein and Painlevé–Gullstrand metrics, and nonstationary Lemaˆıtre–Finkelstein and Kruskal–Szekeres metrics. When the real radial functions of the Dirac equation and of the second-order equation in the Schwarzschild field are used, the domain of wave functions is restricted to the range r > r0, where r0 is the radius of the event horizon. A corresponding constraint also exists in other coordinates for all considered metrics. For the considered metrics, the second-order equations admit the existence of degenerate stationary bound states of fermions with zero energy. As a result, we prove that physically meaningful results for a quantum mechanical description of a particle interaction with a gravitational field are independent of the choice of a solution for the centrally symmetric static gravitational field used.


coordinate transformation Dirac Hamiltonian second-order equation for fermions effective potential degenerate bound state 


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  1. 1.
    K. Schwarzschild, “ Über das Gravitationssfeldeines Massenpunktes nach der Einsteinschen Theorie,” Sitzungsber. Deutsch. Acad. Wissenschaft Berlin Kl. Math. Phys. Tech., 189–196 (1916).zbMATHGoogle Scholar
  2. 2.
    A. S. Eddington, The Mathematical Theory of Relativity, Cambridge Univ. Press, New York (1924).zbMATHGoogle Scholar
  3. 3.
    A. S. Eddington, “A comparison of Whitehead’s and Einstein’s formulae,” Nature, 113, 192 (1924).ADSCrossRefGoogle Scholar
  4. 4.
    D. Finkelstein, “Past–future asymmetry of the gravitational field of a point particle,” Phys. Rev., 110, 965–967 (1958).ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    P. Painlevé, “La mécanique classique et la théorie de la relativité,” C. Roy. Acad. Sci. (Paris), 173, 677–680 (1921).ADSzbMATHGoogle Scholar
  6. 6.
    A. Gullstrand, “Allegemeine lösung des statischen einkörper-problems in der Einsteinshen gravitations theorie,” Arkiv. Mat. Astron. Fys., 16, 1–15 (1922).Google Scholar
  7. 7.
    G. Lemaitre, “L’univers en expansion,” Ann. Soc. Sci. Bruxelles Ser. A, 53, 51–85 (1933).zbMATHGoogle Scholar
  8. 8.
    M. D. Kruskal, “Maximal extension of Schwarzschild metric,” Phys. Rev., 119, 1743–1745 (1960).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    G. Szekeres, “On the singularities of a Riemannian manifold,” Publ. Mat. Debrecen, 7, 285–301 (1960).MathSciNetzbMATHGoogle Scholar
  10. 10.
    N. Deruelle and R. Ruffini, “Quantum and classical relativistic energy states in stationary geometries,” Phys. Lett. B, 52, 437–441 (1974).ADSCrossRefGoogle Scholar
  11. 11.
    T. Damour, N. Deruelle, and R. Ruffini, “On quantum resonances in stationary geometries,” Lett. Nuov. Cim., 15, 257–262 (1976).ADSCrossRefGoogle Scholar
  12. 12.
    I. M. Ternov, V. P. Khalilov, G. A. Chizhov, and A. B. Gaina, “Finite motion of massive particles in Kerr and Schwarzschild fields [in Russian],” Izv. Vuzov. Fizika, No. 9, 109–114 (1978).zbMATHGoogle Scholar
  13. 13.
    A. B. Gaina and G. A. Chizhov, “Radial motion in a Schwarzschild field [in Russian],” Izv. Vuzov. Fizika, 120, No. 4, 120–121 (1980).ADSGoogle Scholar
  14. 14.
    I. M. Ternov, A. B. Gaina, and G. A. Chizhov, “Finite motion of electrons in the field of microscopic black holes,” Sov. Phys. J., 23, 695–700 (1980).CrossRefzbMATHGoogle Scholar
  15. 15.
    D. V. Gal’tsov, G. V. Pomerantseva, and G. A. Chizhov, “Filling of quasibound states by electrons in a schwarzchild field,” Sov. Phys. J., 26, 743–745 (1983).CrossRefGoogle Scholar
  16. 16.
    I. M. Ternov and A. B. Gaina, “Energy spectrum of the Dirac equation in the Schwarzschild and Kerr fields,” Sov. Phys. J., 31, 157–163 (1988).MathSciNetCrossRefGoogle Scholar
  17. 17.
    A. B. Gaina and O. B. Zaslavskii, “On quasilevels in the gravitational field of a black hole,” Class. Q. Grav., 9, 667–676 (1992).ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    A. B. Gaina and N. I. Ionescu-Pallas, “The fine and hyperfine structure of fermionic levels in gravitational fields,” Rom. J. Phys., 38, 729–730 (1993).Google Scholar
  19. 19.
    A. Lasenby, C. Doran, J. Pritchard, A. Caceres, and S. Dolan, “Bound states and decay times of fermions in a Schwarzschild black hole background,” Phys. Rev. D, 72, 105014 (2005); arXiv:gr-qc/0209090v2 (2002).ADSCrossRefGoogle Scholar
  20. 20.
    V. P. Neznamov and I. I. Safronov, “Stationary solutions of second-order equations for point fermions in the Schwarzschild gravitational field,” JETP, 127, 647–658 (2018).ADSCrossRefGoogle Scholar
  21. 21.
    L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics [in Russian], Vol. 3, Quantum Mechanics: Non-Relativistic Theory, Nauka, Moscow (1989); English transl. prev. ed., Pergamon, Oxford (1977).Google Scholar
  22. 22.
    M. V. Gorbatenko, V. P. Neznamov, and E. Yu. Popov, “Analysis of half-spin particle motion in Reissner–Nordström and Schwarzschild fields by the method of effective potentials,” Grav. Cosmol., 23, 245–250 (2017); arXiv:1511.05058v1 [gr-qc] (2015).ADSCrossRefzbMATHGoogle Scholar
  23. 23.
    M. V. Gorbatenko and V. P. Neznamov, “Uniqueness and self-conjugacy of Dirac Hamiltonians in arbitrary gravitational fields,” Phys. Rev. D, 83, 105002 (2011); arXiv:1102.4067v1 [gr-qc] (2011).ADSCrossRefGoogle Scholar
  24. 24.
    J. Schwinger, “Energy and momentum density in field theory,” Phys. Rev., 130, 800–805 (1963).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    M. V. Gorbatenko and V. P. Neznamov, “A modified method for deriving self-conjugate Dirac Hamiltonians in arbitrary gravitational fields and its application to centrally and axially symmetric gravitational fields,” J. Modern Phys., 6, 54289 (2015); arXiv:1107.0844v7 [gr-qc] (2011).CrossRefGoogle Scholar
  26. 26.
    C. M. Bender, D. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett., 89, 270401 (2002); “Extension of PT -symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D, 70, 025001 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys., 43, 205–214 (2002); arXiv:math-ph/0107001v3 (2001); “Pseudo-Hermiticity versus PT-symmetry II: A complete characterization of non-Hermitian Hamiltonians with a real spectrum,” J. Math. Phys., 43, 2814–2816 (2002); arXiv:math-ph/0110016v2 (2001); “Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries,” J. Math. Phys., 43, 3944–3951 (2002); arXiv:math-ph/0203005v2 (2002).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    B. Bagchi and A. Fring, “Minimal length in quantum mechanics and non-Hermitian Hamiltonian systems,” Phys. Lett. A, 373, 4307–4310 (2009).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    L. Parker, “One-electron atom as a probe of spacetime curvature,” Phys. Rev. D, 22, 1922–1934 (1980).ADSCrossRefGoogle Scholar
  30. 30.
    M. V. Gorbatenko and V. P. Neznamov, “Solution of the problem of uniqueness and Hermiticity of Hamiltonians for Dirac particles in gravitational fields,” Phys. Rev. D, 82, 104056 (2010); arXiv:1007.4631v1 [gr-qc] (2010).ADSCrossRefGoogle Scholar
  31. 31.
    D. R. Brill and J. A. Wheeler, “Interaction of neutrinos and gravitational fields,” Rev. Modern Phys., 29, 465–479 (1957).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    S. R. Dolan, “Scattering, absorption, and emission by black holes,” Doctoral dissertation, Univ. of Cambridge, Cambridge (2006).Google Scholar
  33. 33.
    V. P. Neznamov, “Second-order equations for fermions on Schwarzschild, Reissner–Nordström, Kerr, and Kerr–Newman space–times,” Theor. Math. Phys., 197, 1823–1837 (2018).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Russian Federal Nuclear CenterAll-Russian Scientific Research Institute of Experimental PhysicsSarov, Nizhny Novgorod OblastRussia
  2. 2.National Research Nuclear University MEPhIMoscowRussia

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